Probability of direction

In Bayesian statistics, the Probability of Direction (pd) is a measure of effect existence representing the certainty with which an effect is positive or negative. This index is numerically similar to the frequentist p-value.

Definition

It is mathematically defined as the proportion of the posterior distribution that is of the median's sign. It typically varies between 50% and 100%.

Properties

The probability of direction is typically independent from the statistical model, as it is solely based on the posterior distribution and does not require any additional information from the data or the model. Contrary to indices related to the Region of Practical Interest (ROPE), it is robust to the scale of both the response variable and the predictors. Similarly to its frequentist counterpart the p-value, this index is not able to quantify evidence in favor of the null hypothesis.

Relationship with p-value

The probability of direction has a direct correspondence with the frequentist one-sided p-value through the formula $p_{one-sided}=1-pd$ and to the two-sided 'p'-value through the formula $p_{two-sided}=2*(1-pd)$. Thus, a two-sided p-value of respectively .1, .05, .01 and .001 would correspond approximately to a pd of 95%, 97.5%, 99.5% and 99.95%. The proximity between the pd and the p-value follows the original definition of the latter as an index of effect existence rather than significance ("worth of interest").

Interpretation

The 'bayestestR' package for R suggests the following rule of thumb guidelines:

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pd	p-value equivalence	Interpretation
$\leq$ 95%	p > .1	Uncertain
$>$ 95%	p < .1	Possibly existing
$>$ 97%	p $\approx$ .06	Likely existing
$>$ 99%	p $\approx$ .02	Probably existing
$>$ 99.9%	p $\approx$ .002	Certainly existing

See also

• Bayes factor

External links

• bayestestR — an R package for computing Bayesian indices